Arithmetic and Analysis of the series $\sum _{n=1}^{\infty }\frac{1}{n}\sin \frac{x}{n}$

Abstract

In this paper we connect a celebrated theorem of Nyman and Beurling on the equivalence between the Riemann hypothesis and the density of some functional space in $L^2(0,1)$ to a trigonometric series considered first by Hardy and Littlewood. We highlight some of its curious analytical and arithmetical properties.